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EDIT: I would like to make a list of modern applications of algebra in analysis. By "modern" I will mean developments since the beginning of the 20th century. It is well known that classical linear algebra of the 19th century (or earlier), such as vector spaces, determinants, diagonalization and Jordan form of matrices, have many applications in other areas of mathematics and beyond that, in particular in analysis. Below I will give an example of a modern application.

The choice of areas is motivated by my personal taste.

Now let me describe one of my favorite examples. Let $P$ be real a polynomial in $n$ variables. For any smooth compactly supported test function $\phi$ consider the integral $\int_{\mathbb{R}^n}|P(x)|^\lambda\phi(x)dx$. It converges absolutely for $Re(\lambda)>0$ and is holomorphic in $\lambda$ there. The question posed by I.M. Gelfand was whether it has meromoprhic continuation in $\lambda$ to the whole complex plane.

This problem has positive answer. In full generality it was solved by J. Bernstein and S. Gelfand, and independently by M. Atiyah, in 1969, see http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bern-Gel-P-lam-FAN.pdf They used the Hironaka's theorem on resolution of singularities. That is a deep important result from algebraic geometry for which Hironaka was awarded Fields medal.

A little later, in 1972, J. Bernstein invented a different proof of the meromorphic continuation theorem which did not use Hironaka's result. The main step was the following statement. There exists a differential operator $D_\lambda$ on $\mathbb{R}^n$ whose coefficients depend polynomially on $x$ and are rational functions of $\lambda$ such that $$|P|^\lambda=D_\lambda (|P|^{\lambda +1}) \mbox{ for } Re(\lambda) >0.$$ Using this functional equation one can recursively extend the above integral to the whole complex plane.

In order to prove the above functional equation J. Bernstein developed a purely algebraic method which later on became fundamental in the theory of algebraic D-modules.

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    $\begingroup$ math.stackexchange.com/questions/59768/… $\endgroup$ – Carlo Beenakker Feb 1 '17 at 20:19
  • $\begingroup$ Is the algebra in the Atiyah–Singer index theorem modern enough to qualify? $\endgroup$ – user95282 Feb 1 '17 at 20:42
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    $\begingroup$ I'm not sure what you mean by "modern analysis", but in functional analysis, which has to be at least a large chunk of what you mean, theory building clearly predominates. $\endgroup$ – Nik Weaver Feb 1 '17 at 22:06
  • $\begingroup$ @user95282: Atiyah-Singer is modern enough, but I think it uses topology rather than algebra. $\endgroup$ – orbits Feb 2 '17 at 4:49
  • $\begingroup$ @NikWeaver: I did not mean to enter this discussion here, but may be this is just my perception due to my background. Say in modern Banach space theory problem solving predominates. $\endgroup$ – orbits Feb 2 '17 at 6:52
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One famous example for me is the Kyoto school and the "algebraic analysis" area founded by Mikio Sato. Thanks to sheaf theory and the concepts of derived category, Grothendieck's six operations and homological algebra one can do microlocal analysis (notably the notion of micro-support which generalizes the notion of propagation in PDE), study analytic D-modules (the Riemann-Hilbert correspondance is a well known result proved by algebraic methods), have a good cohomological definition of hyperfunctions, etc ...

A good overview of this theory can be found here. However, these methods are very effective to study linear PDE but seems currently unable to deal with non-linear cases.

Let me perhaps quote a beautiful theorem which highlight the link with PDE. This is theorem 11.3.3 in P. Schapira and M. Kashiwara, Sheaves on manifold.

Let $X$ be a complex manifold, M a coherent $D_X$-module and $Sol_X(M)$ be the solution complex of $M$. Then : $$SS(Sol_X(M)) = \text{char}(M).$$

Here, $SS$ is the micro support and $\text{char}$ the characteristic variety. Actually this theorem is a generalization and sheaf abstraction of the following proposition

Let $P$ be a holomorphic differential operator defined on $X$ and $\phi$ be a real $C^1$-function on $X$ such that $\sigma(P)(d\phi_x) \neq 0.$ on a $X$. (Here $\sigma(P)$ denotes the principal symbol) Let $$\Omega = \{x \in X : \phi(x)<0\}$$ and let $f\in \mathcal{O}_X(\Omega)$ be such that $Pf$ extends holomorphically on a neighborhood of $x_0\in \partial \Omega.$ Then $f$ extends holomorphically in a neighborhood of $x_0$.

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  • $\begingroup$ In the final proposition, what do you mean by $(x; \partial\phi(x)$? Is that the exterior derivative at the point $x$ of $\phi$? If so, I think a more common notation would be something like $d\phi_x$. $\endgroup$ – Avi Steiner Feb 2 '17 at 14:37
  • $\begingroup$ Thank you, there is a typo in the book, it is $d\phi_x$. $\endgroup$ – C. Dubussy Feb 2 '17 at 14:54
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Picard-Vessiot theory is an application of Galois theory to the theory of linear differential equations. A major goal is to describe when the differential equation can be solved by quadratures in terms of properties of the differential Galois group.

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Deligne's solution of the Ramanujan conjecture on estimate of coefficients of an automorphic form.

To state it, define for $|q|<1$
$$D(q):=q\prod_{n=1}^\infty(1-q^n)^{24}=\sum_{n=1}^\infty \tau(n)q^n.$$ Then the Ramanujan's conjecture (or Deligne's theorem) says that for any prime number $p$ one has $$|\tau(p)|<2p^{11/2}.$$ Deligne used a surprising interpretation of $\tau(p)$ as a trace of an element (called Frobenius element) of some Galois group in cohomology of an arithmetic variety with coefficients in a sheaf. To get the estimate, Deligne used the Weil conjectures predicting the behavior of eigenvalues of the Frobenius element. (The corresponding statement of the Weil conjectures Deligne proved few year after his work on the Ramanujan conjecture.)

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Khovanskii (among others) has been able to deduce the Aleksandrov-Fenchel inequalities on mixed volumes from the Hodge index theorem. You can check the results and references in this paper.

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    $\begingroup$ There is also an expository paper of Gromov that I like a lot: cims.nyu.edu/~gromov/… $\endgroup$ – alpx Dec 21 '18 at 19:27
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I assume differential geometry counts as analysis to you. After all, $k$th order differential operators taking sections of a bundle $E \to M$ to sections of a bundle $E' \to M$ can be characterized as bundle maps $J^{k} E \to E'$.

The representation theory of Lie groups and Lie algebras plays a central role in differential geometry, especially in parabolic geometry and more generally Cartan geometry. If you browse through Andreas Cap's publications, you'll see how deeply these areas are intertwined.

For example, homomorphisms of Verma modules, BGG resolutions, and Lie algebra cohomology are important tools for understanding parabolic geometries and their invariant differential operators. For details, have a look at this paper and its sequel by Cap and Soucek.

For another example, consider an embedded Lie subgroup $P \hookrightarrow G$ and a $G$-representation $V$. Given a parallel section of a tractor bundle associated to a $G$-rep $V$, we obtain a curved orbit partition (essentially a sort of holonomy reduction) by taking preimages of the $P$-orbits. So it is important to know the $G$ and $P$ orbits in the representation.

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The second theorem below was first proved by Wiener (1933). Later Gelfand (1941) found a wonderful algebraic proof based on Banach algebras, which I included since it is so simple and elegant.

Theorem (Gelfand-Mazur) If $A$ is a complex Banach algebra with unity $e$ whose nonzero elements are invertible, then $A\cong\mathbb{C}$.

Proof (sketch). For any $a\in A$ there exists $\lambda(a)\in\mathbb{C}$ such that $a=\lambda(a)e$: otherwise $a-\lambda e$ would be invertible for any $\lambda\in\mathbb{C}$ and, choosing any $\phi\in A^*$ (the dual space) such that $\phi(a^{-1})\neq 0$, the holomorphic map $\lambda\mapsto\phi((a-\lambda e)^{-1})$ would contradict Liouville's theorem. Clearly $a\mapsto\lambda(a)$ defines an isomorphism with $\mathbb{C}$.

Theorem (Wiener). If $f:S^1\to\mathbb{C}\setminus\{0\}$ has the form $f(e^{i\theta})=\sum_{n\in\mathbb{Z}}c_n e^{in\theta}$ with $\sum|c_n|<\infty$, then $\frac{1}{f}$ has the same form (i.e. $\frac{1}{f(e^{i\theta})}=\sum_{n\in\mathbb{Z}}c_n'e^{in\theta}$ with $\sum|c_n'|<\infty$).

Proof (Gelfand). The set of functions $g(e^{i\theta})=\sum_{n\in\mathbb{Z}}a_n e^{in\theta}$ with $\sum|a_n|<\infty$ forms a commutative Banach algebra $B$ with the norm $\|g\|:=\sum|a_n|$ (and multiplicative identity $1$). The thesis amounts to show that $f$ is invertible in $B$.

If this does not happen, then $f$ is contained in some maximal ideal $M$, which has to be closed by maximality (since invertible elements form an open set). By Gelfand-Mazur theorem $B/M\cong\mathbb{C}$, so there exists a homomorphism $\phi:B\to\mathbb{C}$ satisfying $\phi(f)=0$. Notice that, for any $b\in B$, $|\phi(b)|\le\|b\|$: indeed, whenever $|\lambda|>\|b\|$ the element $b-\lambda\cdot 1=-\lambda(1-\lambda^{-1}b)$ is invertible in $B$ and thus $\phi(b)-\lambda=\phi(b-\lambda\cdot 1)\neq 0$. In particular, $\phi$ is continuous.

Let $h(e^{i\theta}):=e^{i\theta}$. Observe that $h\in B$ is invertible and that, for any $n\in\mathbb{Z}$, $\|h^n\|=1$. So $|\phi(h)|^n=|\phi(h^n)|\le 1$ for all $n$, i.e. $\phi(h)\in S^1$. Thus we arrive at the contradiction $$ 0=\phi(f)=\sum c_n\phi(h^n)=\sum c_n\phi(h)^n=f(\phi(h)). $$

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I don't know very much about this, but Chuu-Lian Terng and Karen Uhlenbeck have used loop groups and some Lie theory to solve certain PDEs: http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_44.pdf

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I'm far from an expert on the subject, but the theory of isoparametric surfaces in the sphere is an interesting example.

Roughly speaking, an isoparametric surface is one whose principal curvatures are all constant counting multiplicities. They are related to questions in PDE as they are the level sets of isoparametric functions (which are relevant to study of wave propagation).

In euclidean space the only (complete) examples are spheres, cylinders and planes (the same is true in Hyperbolic space) however in spheres there are many more examples and it is a famous open question (one of Yau's problems) to classify them.

As I understand it, most of the progress on this problem involves the use of deep algebraic results.

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To add to the example in OP and many of the answers, this expository

may be interesting to read.

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